Mathematical Methods for Physics and Engineering - Matematica.NET

11.8 DIVERGENCE THEOREM AND RELATED THEOREMSSince v has a singularity at the origin it is not differentiable there, i.e. ∇·v is notdefined there, but at all other points ∇ · v = 0, as required for an incompressiblefluid. Therefore, from the divergence theorem, for any closed surface S 2 that doesnot enclose the origin we have∮ ∫v · dS = ∇ · v dV =0.S 2 VThus we see that the surface integral ∮ Sv · dS has value Q or zero dependingon whether or not S encloses the source. In order that the divergence theorem isvalid for all surfaces S, irrespective of whether they enclose the source, we write∇ · v = Qδ(r),where δ(r) is the three-dimensional Dirac delta function. The properties of thisfunction are discussed fully in chapter 13, but for the moment we note that it isdefined in such a way that∫Vδ(r − a) =0 forr ≠ a,{f(a) if a lies in Vf(r)δ(r − a) dV =0 otherwisefor any well-behaved function f(r). Therefore, for any volume V containing thesource at the origin, we have∫∫∇ · v dV = Q δ(r) dV = Q,Vwhich is consistent with ∮ Sv · dS = Q for a closed surface enclosing the source.Hence, by introducing the Dirac delta function the divergence theorem can bemade valid even for non-differentiable point sources.The generalisation to several sources and sinks is straightforward. For example,if a source is located at r = a and a sink at r = b then the velocity field isv =and its divergence is given byV(r − a)Q (r − b)Q−4π|r − a|34π|r − b| 3∇ · v = Qδ(r − a) − Qδ(r − b).Therefore, the integral ∮ Sv · dS has the value Q if S encloses the source, −Q ifS encloses the sink and 0 if S encloses neither the source nor sink or enclosesthem both. This analysis also applies to other physical systems – for example, inelectrostatics we can regard the sources and sinks as positive and negative pointcharges respectively and replace v by the electric field E.405